# Search Results

## You are looking at 1 - 10 of 25 items for

- Author or Editor: Howard W. Barker x

- All content x

## Abstract

The Monte Carlo method of photon transport was used to simulate solar radiative transfer for cumulus-like cloud forms (and cloud fields) possessing structural characteristics similar to those induced by wind shear. Using regular infinite arrays of finite, slanted-cuboidal clouds (parallelepipeds), it was demonstrated that the magnitude of cloud field albedo variation as a function of relative solar azimuth angle (up to 40% of albedo) can be larger than the albedo disparities between plane-parallel clouds and fields of nonsheared finite clouds. In general, cloud field albedo is maximized when shearing is away from the sun and minimized when shearing is toward the sun. This is explained by changes in effective cloud fraction presented to the direct solar beam. The albedo of individual clouds, on the other hand, is maximized when shearing is toward the sun, especially when shearing angle equals solar zenith angle. This is because of both reduced irradiance onto cloud sides and enhanced effective optical depth of cloud. These results were corroborated by conducting similar experiments using realistic cloud forms generated by a dynamical/microphysical cloud model. The magnitude of albedo differences between sheared and corresponding nonsheared broken clouds reached 25% of the albedo. Again, this is due to differing effective cloud fractions and side illumination.

It was found that the bidirectional reflectance functions (BDRFs) of sheared clouds are sensitive to solar azimuth angle. Relative differences between BDRFs for clouds sheared toward and away from the sun can be as large as 50% for arrays of idealized parallelepiped clouds and 25% for more realistic clouds. Differences are minimized when viewing is perpendicular to the wind shear direction provided clouds are sheared toward or away from the sun. BDRFs for sheared clouds are much more asymmetric near the zenith than BDRFs for corresponding cubic (nonsheared) clouds. Hence, viewing sheared clouds at a 60° zenith angle will not necessarily provide least biased estimates of cloud field albedo as is the case for nonsheared clouds. Finally, it was demonstrated that BDRF differences arising from use of Mie and Henyey–Greenstein phase functions are substantially smaller than differences associated with varying solar azimuth angle.

## Abstract

The Monte Carlo method of photon transport was used to simulate solar radiative transfer for cumulus-like cloud forms (and cloud fields) possessing structural characteristics similar to those induced by wind shear. Using regular infinite arrays of finite, slanted-cuboidal clouds (parallelepipeds), it was demonstrated that the magnitude of cloud field albedo variation as a function of relative solar azimuth angle (up to 40% of albedo) can be larger than the albedo disparities between plane-parallel clouds and fields of nonsheared finite clouds. In general, cloud field albedo is maximized when shearing is away from the sun and minimized when shearing is toward the sun. This is explained by changes in effective cloud fraction presented to the direct solar beam. The albedo of individual clouds, on the other hand, is maximized when shearing is toward the sun, especially when shearing angle equals solar zenith angle. This is because of both reduced irradiance onto cloud sides and enhanced effective optical depth of cloud. These results were corroborated by conducting similar experiments using realistic cloud forms generated by a dynamical/microphysical cloud model. The magnitude of albedo differences between sheared and corresponding nonsheared broken clouds reached 25% of the albedo. Again, this is due to differing effective cloud fractions and side illumination.

It was found that the bidirectional reflectance functions (BDRFs) of sheared clouds are sensitive to solar azimuth angle. Relative differences between BDRFs for clouds sheared toward and away from the sun can be as large as 50% for arrays of idealized parallelepiped clouds and 25% for more realistic clouds. Differences are minimized when viewing is perpendicular to the wind shear direction provided clouds are sheared toward or away from the sun. BDRFs for sheared clouds are much more asymmetric near the zenith than BDRFs for corresponding cubic (nonsheared) clouds. Hence, viewing sheared clouds at a 60° zenith angle will not necessarily provide least biased estimates of cloud field albedo as is the case for nonsheared clouds. Finally, it was demonstrated that BDRF differences arising from use of Mie and Henyey–Greenstein phase functions are substantially smaller than differences associated with varying solar azimuth angle.

## Abstract

This study examines the ability to estimate regional cloud albedo using 1D series of cloud optical depth τ similar to those inferred from ground-based microwave radiometers. The investigation has two facets: use of appropriate radiative transfer algorithms and adequate portrayal of cloud structure. Using 1024 × 1024 pixel arrays of τ inferred from 28.5-m resolution Landsat data, regional albedos and albedos along individual scanlines are computed by a 3D Monte Carlo (MC) photon transport algorithm. Assuming the scanlines to be proxies for 1D series of τ a 2D MC algorithm and the Independent Pixel Approximation (IPA) are used to compute albedos for scanlines of various lengths and resolutions.

Regarding the appropriateness of doing radiative transfer calculations on a 1D series of τ, it is shown that for 1D series of τ containing 1024 pixels (∼30 km), lack of information about cloud structure adjacent to the series yields root-mean-square errors for 2D MC albedos of about 2% for a stratocumulus and 20% for two cumulus cloud fields. For the cumulus cases there is a marked tendency to over (under) estimate albedos for relatively bright (dark) scanlines. For the same series, the IPA performs very well relative to the, 2D MC for stratocumulus conditions. For broken cumulus clouds, however, notable biases between the 2D MC and IPA results stem mostly from the neglect of cloud sides by the IPA. In all cases, random errors are small.

The IPA is then used to investigate the accuracy of estimating regional cloud albedo with 1D datasets containing various amounts of information. It is demonstrated that for series with less than 100 pixels (≲3 km) at full resolution, the probability of attaining good estimates of regional cloud albedo is very low regardless of cloud type. Ideally, series with more than 1024 pixels should be used. Regarding sensitivity to data resolution, estimated regional albedos are almost resolution independent for pixel sizes up to about 2 km (∼70 pixels). At coarser resolutions. loss of cloud structure information important for radiative transfer is great and the quality of regional albedo estimates degraded, particularly for oblique sun.

## Abstract

This study examines the ability to estimate regional cloud albedo using 1D series of cloud optical depth τ similar to those inferred from ground-based microwave radiometers. The investigation has two facets: use of appropriate radiative transfer algorithms and adequate portrayal of cloud structure. Using 1024 × 1024 pixel arrays of τ inferred from 28.5-m resolution Landsat data, regional albedos and albedos along individual scanlines are computed by a 3D Monte Carlo (MC) photon transport algorithm. Assuming the scanlines to be proxies for 1D series of τ a 2D MC algorithm and the Independent Pixel Approximation (IPA) are used to compute albedos for scanlines of various lengths and resolutions.

Regarding the appropriateness of doing radiative transfer calculations on a 1D series of τ, it is shown that for 1D series of τ containing 1024 pixels (∼30 km), lack of information about cloud structure adjacent to the series yields root-mean-square errors for 2D MC albedos of about 2% for a stratocumulus and 20% for two cumulus cloud fields. For the cumulus cases there is a marked tendency to over (under) estimate albedos for relatively bright (dark) scanlines. For the same series, the IPA performs very well relative to the, 2D MC for stratocumulus conditions. For broken cumulus clouds, however, notable biases between the 2D MC and IPA results stem mostly from the neglect of cloud sides by the IPA. In all cases, random errors are small.

The IPA is then used to investigate the accuracy of estimating regional cloud albedo with 1D datasets containing various amounts of information. It is demonstrated that for series with less than 100 pixels (≲3 km) at full resolution, the probability of attaining good estimates of regional cloud albedo is very low regardless of cloud type. Ideally, series with more than 1024 pixels should be used. Regarding sensitivity to data resolution, estimated regional albedos are almost resolution independent for pixel sizes up to about 2 km (∼70 pixels). At coarser resolutions. loss of cloud structure information important for radiative transfer is great and the quality of regional albedo estimates degraded, particularly for oblique sun.

## Abstract

A method of computing grid-averaged solar radiative fluxes for horizontally inhomogeneous marine boundary layer cloud fields is presented. Its underlying assumptions are as follows: i) the independent pixel approximation (IPA) is applicable and ii) for regions the size of general circulation model (GCM) grid cells, frequency distributions of cloud optical depth τ can be approximated by gamma distribution functions. Equations are furnished for albedo and transmittance that, when applied to judiciously chosen spectral bands, require about three to four times as much CPU time as plane-parallel, homogeneous (PPH) two-stream approximations, which are ubiquitous to GCMs. This is not a hindrance, as two-stream solutions command typically less than 1% of a GCM's CPU consumption. This method, referred to as the gamma IPA, requires estimates of the mean and variance of τ for each applicable grid cell.

Biases associated with PPH models are assessed assuming that cloud properties in GCMs are tuned to yield albedos that agree with those inferred from satellite data. Thus, it is pertinent to ask: when cloud albedos for the gamma IPA and PPH models are forced to be equal, how do their cloud liquid water paths L, droplet effective radii *r _{e}*, and droplet absorptances differ? When albedos are equalized by altering ℒ (fixed

*r*), absorptance differences are generally within ±5%, but values of ℒ for the IPA exceed those for the PPH model often by much more than 20%, depending on ℒ and the extent of inhomogeneity. On the other hand, alteration of

_{e}*r*, (fixed ℒ) requires that the IPA use smaller values of

_{e}*r*than the PPR model. Therefore, since droplet single-scattering albedos increase with decreasing

_{e}*r*, IPA absorptances are generally 5%–50% less than PPH absorptances, depending on ℒ and the extent of inhomogeneity. The overall implications are that by representing subgrid variability of marine boundary layer clouds in GCMs i) ℒ will increase, ii)

_{e}*r*will decrease, and iii) there will probably he slightly less solar absorption by clouds relative to current values. Moreover, the magnitude of absorptance differences depend in part on the number of spectral bands

_{e}*J*used to resolve the solar spectrum. In general, differences for

*J*= 4 and

*J*= 24 are approximately equivalent but for

*J*<4, as in most GCMs, absorptance differences between the gamma IPA and PPH models are exaggerated and often of the wrong sign relative to those for

*J*= 24.

## Abstract

A method of computing grid-averaged solar radiative fluxes for horizontally inhomogeneous marine boundary layer cloud fields is presented. Its underlying assumptions are as follows: i) the independent pixel approximation (IPA) is applicable and ii) for regions the size of general circulation model (GCM) grid cells, frequency distributions of cloud optical depth τ can be approximated by gamma distribution functions. Equations are furnished for albedo and transmittance that, when applied to judiciously chosen spectral bands, require about three to four times as much CPU time as plane-parallel, homogeneous (PPH) two-stream approximations, which are ubiquitous to GCMs. This is not a hindrance, as two-stream solutions command typically less than 1% of a GCM's CPU consumption. This method, referred to as the gamma IPA, requires estimates of the mean and variance of τ for each applicable grid cell.

Biases associated with PPH models are assessed assuming that cloud properties in GCMs are tuned to yield albedos that agree with those inferred from satellite data. Thus, it is pertinent to ask: when cloud albedos for the gamma IPA and PPH models are forced to be equal, how do their cloud liquid water paths L, droplet effective radii *r _{e}*, and droplet absorptances differ? When albedos are equalized by altering ℒ (fixed

*r*), absorptance differences are generally within ±5%, but values of ℒ for the IPA exceed those for the PPH model often by much more than 20%, depending on ℒ and the extent of inhomogeneity. On the other hand, alteration of

_{e}*r*, (fixed ℒ) requires that the IPA use smaller values of

_{e}*r*than the PPR model. Therefore, since droplet single-scattering albedos increase with decreasing

_{e}*r*, IPA absorptances are generally 5%–50% less than PPH absorptances, depending on ℒ and the extent of inhomogeneity. The overall implications are that by representing subgrid variability of marine boundary layer clouds in GCMs i) ℒ will increase, ii)

_{e}*r*will decrease, and iii) there will probably he slightly less solar absorption by clouds relative to current values. Moreover, the magnitude of absorptance differences depend in part on the number of spectral bands

_{e}*J*used to resolve the solar spectrum. In general, differences for

*J*= 4 and

*J*= 24 are approximately equivalent but for

*J*<4, as in most GCMs, absorptance differences between the gamma IPA and PPH models are exaggerated and often of the wrong sign relative to those for

*J*= 24.

## Abstract

In an important paper written over 25 years ago, solar radiative fluxes based on exact Mie phase functions were compared to fluxes for corresponding elliptic and Henyey–Greenstein scattering phase functions. The poor performance of the elliptic function can be attributed to the method used to define its parameter. In this paper, a method is given that yields more appropriate elliptic phase functions. This greatly improves the credibility of the elliptic phase function as a potential candidate for use in two-stream approximations. This is especially true for aerosol conditions where it appears to be more suitable than the Henyey-Greenstein phase function.

## Abstract

In an important paper written over 25 years ago, solar radiative fluxes based on exact Mie phase functions were compared to fluxes for corresponding elliptic and Henyey–Greenstein scattering phase functions. The poor performance of the elliptic function can be attributed to the method used to define its parameter. In this paper, a method is given that yields more appropriate elliptic phase functions. This greatly improves the credibility of the elliptic phase function as a potential candidate for use in two-stream approximations. This is especially true for aerosol conditions where it appears to be more suitable than the Henyey-Greenstein phase function.

## Abstract

It has been hypothesized that over the past ∼200 years, industrial activity has enhanced the number of cloud condensation nuclei (CCN) in the lower atmosphere thereby reducing cloud droplet effective radii *r*
_{e} and increasing the albedo of clouds. It is thought that in some regions, cloud albedos have increased so much that they have greatly ameliorated coincidental forcing by increased concentrations of greenhouse gases. The best estimates of this ameliorating effect come from large-scale climate/chemical transport models that assume clouds to be horizontally homogeneous at scales smaller than several hundred kilometers. It is demonstrated here that for a 2-*μ*m reduction in *r*
_{e}, conventional estimates of increased cloud albedo due to more CCN may be too large by up to, and possibly exceeding, 50%. The largest overestimates occur when reductions to *r*
_{e} are accompanied by enhancements to both cloud variability and liquid water paths. This is attributed to fundamental differences in the way homogeneous and inhomogeneous clouds transport solar radiation.

## Abstract

It has been hypothesized that over the past ∼200 years, industrial activity has enhanced the number of cloud condensation nuclei (CCN) in the lower atmosphere thereby reducing cloud droplet effective radii *r*
_{e} and increasing the albedo of clouds. It is thought that in some regions, cloud albedos have increased so much that they have greatly ameliorated coincidental forcing by increased concentrations of greenhouse gases. The best estimates of this ameliorating effect come from large-scale climate/chemical transport models that assume clouds to be horizontally homogeneous at scales smaller than several hundred kilometers. It is demonstrated here that for a 2-*μ*m reduction in *r*
_{e}, conventional estimates of increased cloud albedo due to more CCN may be too large by up to, and possibly exceeding, 50%. The largest overestimates occur when reductions to *r*
_{e} are accompanied by enhancements to both cloud variability and liquid water paths. This is attributed to fundamental differences in the way homogeneous and inhomogeneous clouds transport solar radiation.

## Abstract

A method for inferring cloud optical depth *τ* is introduced and assessed using simulated surface radiometric measurements produced by a Monte Carlo algorithm acting on fields of broken, single-layer, boundary layer clouds derived from Landsat imagery. The method utilizes a 1D radiative transfer model and time series of zenith radiances and irradiances measured at two wavelengths, *λ*
_{1} and *λ*
_{2}, from a single site with surface albedos *α*_{λ1}*α*_{λ2}*τ*′ are obtained through cloud-base reflectances that are approximated by differencing spectral radiances and estimating upwelling fluxes at cloud base. When initialized with suitable values of *α*_{λ1}*α*_{λ2}*h,* this method performs well at all solar zenith angles. Relative mean bias errors for *τ*′ are typically less than 5% for these cases. Relative variances for *τ*′ for given values of inherent *τ* are almost independent of inherent *τ* and are <50%. Errors due to neglect of net horizontal transport in clouds yield slight, but systematic, overestimates for *τ* ≲ 5 and underestimates for larger *τ.* Frequency distributions and power spectra for retrieved and inherent *τ* are often in excellent agreement. Estimates of *τ* depend weakly on errors in *h,* especially when *h* is overestimated. Also, they are almost insensitive to errors in surface albedo when *α*_{λ1}*α*_{λ2}*τ,* particularly large *τ.* In contrast, the conventional method of using only surface irradiance yields almost entirely invalid results when clouds are broken.

Though results are shown only for surfaces resembling green vegetation (i.e., *α*_{λ1}*α*_{λ2}*α*_{λ1}*α*_{λ2}*τ* for broken clouds above many surface types.

## Abstract

A method for inferring cloud optical depth *τ* is introduced and assessed using simulated surface radiometric measurements produced by a Monte Carlo algorithm acting on fields of broken, single-layer, boundary layer clouds derived from Landsat imagery. The method utilizes a 1D radiative transfer model and time series of zenith radiances and irradiances measured at two wavelengths, *λ*
_{1} and *λ*
_{2}, from a single site with surface albedos *α*_{λ1}*α*_{λ2}*τ*′ are obtained through cloud-base reflectances that are approximated by differencing spectral radiances and estimating upwelling fluxes at cloud base. When initialized with suitable values of *α*_{λ1}*α*_{λ2}*h,* this method performs well at all solar zenith angles. Relative mean bias errors for *τ*′ are typically less than 5% for these cases. Relative variances for *τ*′ for given values of inherent *τ* are almost independent of inherent *τ* and are <50%. Errors due to neglect of net horizontal transport in clouds yield slight, but systematic, overestimates for *τ* ≲ 5 and underestimates for larger *τ.* Frequency distributions and power spectra for retrieved and inherent *τ* are often in excellent agreement. Estimates of *τ* depend weakly on errors in *h,* especially when *h* is overestimated. Also, they are almost insensitive to errors in surface albedo when *α*_{λ1}*α*_{λ2}*τ,* particularly large *τ.* In contrast, the conventional method of using only surface irradiance yields almost entirely invalid results when clouds are broken.

Though results are shown only for surfaces resembling green vegetation (i.e., *α*_{λ1}*α*_{λ2}*α*_{λ1}*α*_{λ2}*τ* for broken clouds above many surface types.

## Abstract

The two primary foci of this note are to assess the ability of the multilayer gamma-weighted two-stream approximation (GWTSA) to compute domain-averaged solar radiative fluxes and to demonstrate how its execution time can be reduced with negligible impact on performance. In addition to the usual parameters needed by a 1D solar code, the GWTSA requires *ν* ∈ R^{+}, which depends on both the horizontal mean and mean logarithm of cloud water content. Reduced central processing unit (CPU) time is realized by simply rounding *ν* to the nearest whole number, denoted as [*ν*]. The experiment reported on here uses 120 fields generated by a 2D cloud-resolving model simulation of an evolving tropical mesoscale convective cloud system. Benchmark calculations are provided by the independent column approximation (ICA), and results are also shown for the conventional two-stream model.

The full GWTSA yields time- and domain-averaged broadband top-of-atmosphere albedo and surface absorptance values of 0.32 and 0.49, which are very close to the ICA values of 0.32 and 0.47. Correspondingly, the GWTSA using [*ν*] produces 0.34 and 0.46. In contrast, the conventional two-stream’s estimates are 0.56 and 0.20. While mean heating rate errors for the conventional two-stream average about −0.5 K day^{−1} near the surface and almost +2 K day^{−1} at 10 km, they are diminished at both altitudes to ∼0.25 K day^{−1} for the GWTSA regardless of whether *ν* or [*ν*] is used. For this simulation, the GWTSA using [*ν*] requires just ∼25% more CPU time than the conventional two-stream approximation.

## Abstract

The two primary foci of this note are to assess the ability of the multilayer gamma-weighted two-stream approximation (GWTSA) to compute domain-averaged solar radiative fluxes and to demonstrate how its execution time can be reduced with negligible impact on performance. In addition to the usual parameters needed by a 1D solar code, the GWTSA requires *ν* ∈ R^{+}, which depends on both the horizontal mean and mean logarithm of cloud water content. Reduced central processing unit (CPU) time is realized by simply rounding *ν* to the nearest whole number, denoted as [*ν*]. The experiment reported on here uses 120 fields generated by a 2D cloud-resolving model simulation of an evolving tropical mesoscale convective cloud system. Benchmark calculations are provided by the independent column approximation (ICA), and results are also shown for the conventional two-stream model.

The full GWTSA yields time- and domain-averaged broadband top-of-atmosphere albedo and surface absorptance values of 0.32 and 0.49, which are very close to the ICA values of 0.32 and 0.47. Correspondingly, the GWTSA using [*ν*] produces 0.34 and 0.46. In contrast, the conventional two-stream’s estimates are 0.56 and 0.20. While mean heating rate errors for the conventional two-stream average about −0.5 K day^{−1} near the surface and almost +2 K day^{−1} at 10 km, they are diminished at both altitudes to ∼0.25 K day^{−1} for the GWTSA regardless of whether *ν* or [*ν*] is used. For this simulation, the GWTSA using [*ν*] requires just ∼25% more CPU time than the conventional two-stream approximation.

## Abstract

A four-stream solution of the longwave radiative transfer is proposed. It is based on the exact perturbation method utilizing the absorption approximation equation as the zero-order solution. Scattering is handled by the first-order perturbation equation. The two- and four-stream approximations are compared both offline and using data from *CALIPSO*’s dual-wavelength lidar.

## Abstract

A four-stream solution of the longwave radiative transfer is proposed. It is based on the exact perturbation method utilizing the absorption approximation equation as the zero-order solution. Scattering is handled by the first-order perturbation equation. The two- and four-stream approximations are compared both offline and using data from *CALIPSO*’s dual-wavelength lidar.

## Abstract

The disposition of mean July clear-sky solar radiation in the Canadian Climate Centre second-generation general circulation model (CCC-GCMII) was analyzed by comparing top of the atmosphere (TOA) net fluxes with earth radiation budget experiment (ERBE) data and atmospheric and surface net fluxes with values inferred from Li's algorithm using ERBE data and European Centre for Medium-Range Weather Forecasts precipitable water data. GCMII tended to reflect ˜5 W m^{−2} too much to space. Corresponding atmospheric and surface absorption, however, tended to be too low and high, respectively, by ˜30 W m^{−2} over much of the Northern Hemisphere. These results were echoed when GCMII atmospheric absorption was compared to estimated results from Li's algorithm driven by GCMII TOA albedo and precipitable water.

The latest version of the CCC-GCM (GCMIII) has numerous upgrades to its clear-sky solar radiative transfer algorithm, the most important of which involve water vapor transmittances and aerosols that tend to enhance atmospheric absorptance. GCMIII's water vapor transmittance functions derive from Geophysical Fluid Dynamics Laboratory line-by-line results, whereas GCMII's were based on Air Force Geophysical Laboratory data. GCMIII includes crude distributions of background tropospheric aerosols, whereas GCMII neglected aerosols.

Li's algorithm was then driven by GCMIII data, and atmospheric absorption of solar radiation by GCMIII was assessed. Differences between GCMIII's and Li's atmospheric absorption over land were almost always within 5 W m^{−2}. Over oceans, differences were mostly between −5 W m^{−2} and −15 W m^{−2}. This apparent underestimation over GCMIII's oceans probably stems from the algorithm's use of a thin, highly absorbing aerosol.

## Abstract

The disposition of mean July clear-sky solar radiation in the Canadian Climate Centre second-generation general circulation model (CCC-GCMII) was analyzed by comparing top of the atmosphere (TOA) net fluxes with earth radiation budget experiment (ERBE) data and atmospheric and surface net fluxes with values inferred from Li's algorithm using ERBE data and European Centre for Medium-Range Weather Forecasts precipitable water data. GCMII tended to reflect ˜5 W m^{−2} too much to space. Corresponding atmospheric and surface absorption, however, tended to be too low and high, respectively, by ˜30 W m^{−2} over much of the Northern Hemisphere. These results were echoed when GCMII atmospheric absorption was compared to estimated results from Li's algorithm driven by GCMII TOA albedo and precipitable water.

The latest version of the CCC-GCM (GCMIII) has numerous upgrades to its clear-sky solar radiative transfer algorithm, the most important of which involve water vapor transmittances and aerosols that tend to enhance atmospheric absorptance. GCMIII's water vapor transmittance functions derive from Geophysical Fluid Dynamics Laboratory line-by-line results, whereas GCMII's were based on Air Force Geophysical Laboratory data. GCMIII includes crude distributions of background tropospheric aerosols, whereas GCMII neglected aerosols.

Li's algorithm was then driven by GCMIII data, and atmospheric absorption of solar radiation by GCMIII was assessed. Differences between GCMIII's and Li's atmospheric absorption over land were almost always within 5 W m^{−2}. Over oceans, differences were mostly between −5 W m^{−2} and −15 W m^{−2}. This apparent underestimation over GCMIII's oceans probably stems from the algorithm's use of a thin, highly absorbing aerosol.

## Abstract

Optical depths τ_{pp} for broken, shallow clouds over ocean were inferred from Landsat cloud reflectances *R*
_{cld} (0.83 μm) with horizontal resolution of 28.5 m. The values τ_{pp} were obtained by applying an inverse, homogeneous, plane-parallel radiance model to each pixel value of *R*
_{cld}. The primary objective of this study was to estimate optical depth errors incurred by the homogeneous, plane-parallel, independent pixel paradigm. This was achieved by computing reflectances *R*
_{mc} with a 3D Monte Carlo photon transport algorithm that employed τ_{pp} and cloud geometric thicknesses *h* > 0.

A single cloud was isolated for study in which the solar zenith angle was 30° and average τ_{pp} was 5.8. This cloud measured about 1.2 km in diameter but *h* had to be estimated. In the Monte Carlo simulations, *h* was set to be uniform for the entire cloud. For *h* between 150 and 300 m, cloud-average reflectance *R*_{mc}*R*_{cld}_{pp}
^{1/δ(h)} in the Monte Carlo algorithm yielded *R*_{mc}*R*_{cld}*h* = 225 m, 1/δ(*h* = 225) ≈ 1.11, and this increased average τ_{pp} to ∼8.0, which was a 35% increase. At the pixel level, however, random errors associated with fields of *R*
_{mc} − *R*
_{cld} were reduced only slightly when τ_{pp}
^{1/δ(h)} was used rather than τ_{pp}. Finally, τ_{pp}
^{1/δ(h)} was applied to numerous neighboring clouds. When the aspect (height to width) ratio *A* of neighboring clouds was assumed to be constant, τ_{pp} for each cloud received a unique scaling, and this yielded Landsat mean reflectances to within 4% for *A* < 0.3. This suggested that grid-averaged τ_{pp} was likely about 4 rather than 3, as was the plane-parallel, independent pixel estimate.

## Abstract

Optical depths τ_{pp} for broken, shallow clouds over ocean were inferred from Landsat cloud reflectances *R*
_{cld} (0.83 μm) with horizontal resolution of 28.5 m. The values τ_{pp} were obtained by applying an inverse, homogeneous, plane-parallel radiance model to each pixel value of *R*
_{cld}. The primary objective of this study was to estimate optical depth errors incurred by the homogeneous, plane-parallel, independent pixel paradigm. This was achieved by computing reflectances *R*
_{mc} with a 3D Monte Carlo photon transport algorithm that employed τ_{pp} and cloud geometric thicknesses *h* > 0.

A single cloud was isolated for study in which the solar zenith angle was 30° and average τ_{pp} was 5.8. This cloud measured about 1.2 km in diameter but *h* had to be estimated. In the Monte Carlo simulations, *h* was set to be uniform for the entire cloud. For *h* between 150 and 300 m, cloud-average reflectance *R*_{mc}*R*_{cld}_{pp}
^{1/δ(h)} in the Monte Carlo algorithm yielded *R*_{mc}*R*_{cld}*h* = 225 m, 1/δ(*h* = 225) ≈ 1.11, and this increased average τ_{pp} to ∼8.0, which was a 35% increase. At the pixel level, however, random errors associated with fields of *R*
_{mc} − *R*
_{cld} were reduced only slightly when τ_{pp}
^{1/δ(h)} was used rather than τ_{pp}. Finally, τ_{pp}
^{1/δ(h)} was applied to numerous neighboring clouds. When the aspect (height to width) ratio *A* of neighboring clouds was assumed to be constant, τ_{pp} for each cloud received a unique scaling, and this yielded Landsat mean reflectances to within 4% for *A* < 0.3. This suggested that grid-averaged τ_{pp} was likely about 4 rather than 3, as was the plane-parallel, independent pixel estimate.